1. Field of the Invention
The present invention relates to a shape calculation method for calculating a shape of a test object which has an aspheric surface with high accuracy.
2. Description of the Related Art
Japanese Patent Application Laid-Open Nos. 02-259509 and 2004-125768 discuss a method referred to as a “wave field synthesis method” or a “stitching method” as a technique for measuring a shape of a surface of an optical element. The stitching method is described hereinafter with reference to FIGS. 11 and 12.
FIG. 11 illustrates a test surface 4a of an optical element that is a test object, which is seen from above. An abscissa axis is set as an x-axis, while an ordinate axis is set as a y-axis. A test surface is divided into a plurality of measurement regions (sub-apertures), each of which is represented by a small circle and overlaps with other measurement regions. A measurement region wps1 includes a center of the test surface 4a represented as a shaded area. Another measurement region wps2 is represented by a dashed circle and adjoins most closely in a negative y-direction to the measurement region wps1. According to the stitching method, a test surface is divided into a plurality of measurement regions. Then, all measurement data are combined with one another. Thus, the shape of the entire test surface is obtained. The measurement of each measurement region is performed at a position where interference fringes are null fringes. The word “null” indicates a state in which a density of interference fringes is low. More specifically, according to the present invention, the word “null” indicates a state in which a number of interference fringes is 1 or less.
Next, a typical configuration of an apparatus for performing a stitching method is described hereinafter with reference to FIG. 12. An apparatus for measuring a shape of a test surface 14a of a test object 14 includes a light source 11, a half mirror 12, a tilt/shift (TS) lens 13, a reference surface 13a, a holder 15 for holding the test object 14, a stage 16 for driving the test object 14 together with the holder 15, an imaging lens 17, and an image-pickup unit 18. In FIG. 12, a z-axis extends in a lateral direction in a plane of paper, on which FIG. 12 is drawn, in parallel with an optical axis of the TS lens, an x-axis extends in a direction perpendicular to the plane of paper, and a y-axis extends in an up-down direction in the plane of paper. The stage 16 is constituted of a five-axis coordinate system including an x-axis, a y-axis, a z-axis, a θx-axis, and a θz-axis, or a six-axis coordinate system including a θy-axis in addition to the five axes. The θx-axis turns around the x-axis. The θy-axis turns around the y-axis. The θz-axis turns around the z-axis. A shape of the test surface 14a is a sphere.
A light flux which is emitted from the light source 11 is transmitted through the half mirror 12 and incident on the TS lens 13. Thus, a spherical wave is generated. The reference surface 13a serves to generate test light by transmitting a part of the light flux and to generate reference light by reflecting a part of the light flux. The test light transmitted through the reference surface 13a is reflected by the test surface 14a and interferes with the reference light reflected by the reference surface 13a. The interfering light is transmitted again through the TS lens 13 and reflected by the half mirror 12. Then, the reflected light is condensed by the imaging lens 17 to the image-pickup unit 18. Thus, the interfering light is imaged as interference fringes.
In FIG. 12, a first measurement position wp1 is a position of the test surface at which the interference fringes are null fringes when data is measured in the measurement region wps1 which includes a central portion of the test surface 14a illustrated in FIG. 11. When the test object 14 is placed at the first measurement position wp1, the image-pickup unit 18 is placed at a first position cp1 on which the image-pickup unit 18 is conjugate with the test surface 14a. A distance (cavity length) between the reference surface 13a and the test surface 14a at that time is designated by “L1”.
A second measurement position wp2 is another position of the test surface, at which the interference fringes are null fringes when data is measured in the measurement region wps2 illustrated in FIG. 11. When the test object 14 is placed at the second measurement position wp2, the image-pickup unit 18 is placed at a second position cp2 on which the image-pickup unit 18 is conjugate with the test surface 14a. The distance (cavity length) between the reference surface 13a and the test surface 14a at that time is designated by “L2”.
When the test surface is a spherical surface, the cavity length is such that L1=L2. Thus, the position of the image-pickup unit 18 is set so that cp1=cp2. It is unnecessary to adjust the positions of the test surface 14a and the image-pickup unit 18. At all sub-apertures that are not limited to the measurement regions wps1 and wps2 in FIG. 11, the cavity length is equal to L1. Thus, the position of the image-pickup unit 18 is equal to cp1. Consequently, data can be measured in all measurement regions while the position of the image-pickup unit 18 is fixed.
After the shape of each measurement region is calculated by correcting errors at each measurement region, the shapes of all of the measurement regions are joined together. The errors at each measurement region are, e.g., an attitude error, a positioning error, and an abscissa distortion of the test surface and a shape error of the reference surface. An amount of each error is calculated by utilizing an overlapping area of each sub-aperture. Alternatively, the amount of each error is preliminarily and separately measured or calculated and the preliminarily measured or calculated amount of each error is used.
However, if the test surface has an aspheric surface, the interference fringes are not null fringes even when the test surface is illuminated with spherical wave light. Thus, generally, measurement is performed at a position where the number of interference fringes is minimized. In this case, the cavity length varies with the measurement regions. Accordingly, it is necessary to adjust focusing relationship between the test surface and the image-pickup unit. When the focusing relationship between the test surface and the image-pickup unit is adjusted, the focusing relationship between the image-pickup unit and the reference surface changes. Thus, the shape of the reference surface viewed from each measurement region varies. Consequently, more particularly, a high-frequency error occurs in the shape of the reference surface which is calculated using data obtained from a plurality of measurement regions that differ from one another in the focusing relationship. Accordingly, a high-frequency error occurs in the shape of the test surface which is obtained by a difference between data measured by the image-pickup unit and the calculated shape of the reference surface.